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§10.49 Explicit Formulas

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§10.49(i) Unmodified Functions

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Notes:
When \nu=n+\tfrac{1}{2} the asymptotic expansions (10.17.3)–(10.17.6) terminate, and as a consequence of the error bounds of §10.17(iv) they represent the left-hand sides exactly.
Keywords:
spherical Bessel functions
Referenced by:
‣ §10.16, §10.49(ii), §10.61(v), §10.74(i), §11.4(i)
Permalink:
http://dlmf.nist.gov/10.49.i

Define a_{k}(\nu) as in (10.17.1). Then

10.49.1 a_{k}(n+\tfrac{1}{2})=\begin{cases}\dfrac{(n+k)!}{2^{k}k!(n-k)!},&k=0,1,\ldots,n,\\ 0,&k=n+1,n+2,\ldots.\end{cases}
10.49.2 \mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)=\mathop{\sin\/}\nolimits\!\left(z-\tfrac{1}{2}n\pi\right)\sum _{{k=0}}^{{\left\lfloor n/2\right\rfloor}}(-1)^{k}\frac{a_{{2k}}(n+\tfrac{1}{2})}{z^{{2k+1}}}+\mathop{\cos\/}\nolimits\!\left(z-\tfrac{1}{2}n\pi\right)\sum _{{k=0}}^{{\left\lfloor(n-1)/2\right\rfloor}}(-1)^{k}\frac{a_{{2k+1}}(n+\tfrac{1}{2})}{z^{{2k+2}}}.
10.49.3
\mathop{\mathsf{j}_{{0}}\/}\nolimits\!\left(z\right)=\frac{\mathop{\sin\/}\nolimits z}{z},
\mathop{\mathsf{j}_{{1}}\/}\nolimits\!\left(z\right)=\frac{\mathop{\sin\/}\nolimits z}{z^{2}}-\frac{\mathop{\cos\/}\nolimits z}{z},
\mathop{\mathsf{j}_{{2}}\/}\nolimits\!\left(z\right)=\left(-\frac{1}{z}+\frac{3}{z^{3}}\right)\mathop{\sin\/}\nolimits z-\frac{3}{z^{2}}\mathop{\cos\/}\nolimits z.
10.49.4 \mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)=-\mathop{\cos\/}\nolimits\!\left(z-\tfrac{1}{2}n\pi\right)\sum _{{k=0}}^{{\left\lfloor n/2\right\rfloor}}(-1)^{k}\frac{a_{{2k}}(n+\tfrac{1}{2})}{z^{{2k+1}}}+\mathop{\sin\/}\nolimits\!\left(z-\tfrac{1}{2}n\pi\right)\sum _{{k=0}}^{{\left\lfloor(n-1)/2\right\rfloor}}(-1)^{k}\frac{a_{{2k+1}}(n+\tfrac{1}{2})}{z^{{2k+2}}}.
10.49.5
\mathop{\mathsf{y}_{{0}}\/}\nolimits\!\left(z\right)=-\frac{\mathop{\cos\/}\nolimits z}{z},
\mathop{\mathsf{y}_{{1}}\/}\nolimits\!\left(z\right)=-\frac{\mathop{\cos\/}\nolimits z}{z^{2}}-\frac{\mathop{\sin\/}\nolimits z}{z},
\mathop{\mathsf{y}_{{2}}\/}\nolimits\!\left(z\right)=\left(\frac{1}{z}-\frac{3}{z^{3}}\right)\mathop{\cos\/}\nolimits z-\frac{3}{z^{2}}\mathop{\sin\/}\nolimits z.

§10.49(ii) Modified Functions

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Notes:
Use the same method as for §10.49(i), or combine the results of §10.49(i) with (10.47.12) and (10.47.13).
Keywords:
Bessel polynomials, spherical Bessel functions
Referenced by:
‣ §10.39, §10.74(i), §11.4(i), §18.34(i), §18.34(iii), §3.5(viii)
Permalink:
http://dlmf.nist.gov/10.49.ii

Again, with a_{k}(n+\tfrac{1}{2}) as in (10.49.1),

10.49.8 \mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}e^{z}\sum _{{k=0}}^{n}(-1)^{k}\frac{a_{k}(n+\frac{1}{2})}{z^{{k+1}}}+(-1)^{{n+1}}\*\tfrac{1}{2}e^{{-z}}\sum _{{k=0}}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{{k+1}}}.
10.49.9
\mathop{{\mathsf{i}^{{(1)}}_{{0}}}\/}\nolimits\!\left(z\right)=\frac{\mathop{\sinh\/}\nolimits z}{z},
\mathop{{\mathsf{i}^{{(1)}}_{{1}}}\/}\nolimits\!\left(z\right)=-\frac{\mathop{\sinh\/}\nolimits z}{z^{2}}+\frac{\mathop{\cosh\/}\nolimits z}{z},
\mathop{{\mathsf{i}^{{(1)}}_{{2}}}\/}\nolimits\!\left(z\right)=\left(\frac{1}{z}+\frac{3}{z^{3}}\right)\mathop{\sinh\/}\nolimits z-\frac{3}{z^{2}}\mathop{\cosh\/}\nolimits z.
10.49.10 \mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}e^{z}\sum _{{k=0}}^{n}(-1)^{k}\frac{a_{k}(n+\frac{1}{2})}{z^{{k+1}}}+(-1)^{n}\tfrac{1}{2}e^{{-z}}\sum _{{k=0}}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{{k+1}}}.
10.49.11
\mathop{{\mathsf{i}^{{(2)}}_{{0}}}\/}\nolimits\!\left(z\right)=\frac{\mathop{\cosh\/}\nolimits z}{z},
\mathop{{\mathsf{i}^{{(2)}}_{{1}}}\/}\nolimits\!\left(z\right)=-\frac{\mathop{\cosh\/}\nolimits z}{z^{2}}+\frac{\mathop{\sinh\/}\nolimits z}{z},
\mathop{{\mathsf{i}^{{(2)}}_{{2}}}\/}\nolimits\!\left(z\right)=\left(\frac{1}{z}+\frac{3}{z^{3}}\right)\mathop{\cosh\/}\nolimits z-\frac{3}{z^{2}}\mathop{\sinh\/}\nolimits z.
10.49.12 \mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}\pi e^{{-z}}\sum _{{k=0}}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{{k+1}}}.
10.49.13
\mathop{\mathsf{k}_{{0}}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}\pi\frac{e^{{-z}}}{z},
\mathop{\mathsf{k}_{{1}}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}\pi e^{{-z}}\left(\frac{1}{z}+\frac{1}{z^{2}}\right),
\mathop{\mathsf{k}_{{2}}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}\pi e^{{-z}}\left(\frac{1}{z}+\frac{3}{z^{2}}+\frac{3}{z^{3}}\right).

\sum _{{k=0}}^{n}a_{k}(n+\tfrac{1}{2})z^{{n-k}} is sometimes called the Bessel polynomial of degree n. For a survey of properties of these polynomials and their generalizations see Grosswald (1978). See also §18.34, de Bruin et al. (1981a, b), and Dunster (2001c).

§10.49(iv) Sums or Differences of Squares

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Notes:
From (10.18.6), (10.47.3), and (10.47.4), {\mathop{\mathsf{j}_{{n}}\/}\nolimits^{{2}}}\!\left(z\right)+{\mathop{\mathsf{y}_{{n}}\/}\nolimits^{{2}}}\!\left(z\right)=(\pi/(2z)){\mathop{M_{{n+\frac{1}{2}}}\/}\nolimits^{{2}}}\!\left(z\right). Then apply (10.18.17). To derive (10.49.20) combine (10.47.12) and (10.49.18).
Keywords:
spherical Bessel functions
Permalink:
http://dlmf.nist.gov/10.49.iv
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